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受欢迎的三角函数 >

cos^6(x)+3cos^3(x)-4=0

解答

cos^{6} (x) + 3 cos^{3} (x) - 4 = 0

解答

x = 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n

求解步骤

cos^{6} (x) + 3 cos^{3} (x) - 4 = 0

用替代法求解

cos^{6} (x) + 3 cos^{3} (x) - 4 = 0

令

: cos (x) = u

u^{6} + 3 u^{3} - 4 = 0

u^{6} + 3 u^{3} - 4 = 0 : u = 1, u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}, u = - 34, u = \frac{3 4}{2} - i \frac{3 4 3}{2}, u = \frac{3 4}{2} + i \frac{3 4 3}{2}

u^{6} + 3 u^{3} - 4 = 0

用

v = u^{3}

和

v^{2} = u^{6}

改写方程式

v^{2} + 3 v - 4 = 0

解

v^{2} + 3 v - 4 = 0 : v = 1, v = - 4

v^{2} + 3 v - 4 = 0

使用求根公式求解

v^{2} + 3 v - 4 = 0

二次方程求根公式:

若

a = 1, b = 3, c = - 4

v_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot ( - 4 )}{2 \cdot 1}

v_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot ( - 4 )}{2 \cdot 1}

3^{2} - 4 \cdot 1 \cdot (- 4) = 5

3^{2} - 4 \cdot 1 \cdot (- 4)

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 1 \cdot 4

数字相乘:

4 \cdot 1 \cdot 4 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

v_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 1}

将解分隔开

v_{1} = \frac{- 3 + 5}{2 \cdot 1}, v_{2} = \frac{- 3 - 5}{2 \cdot 1}

v = \frac{- 3 + 5}{2 \cdot 1} : 1

\frac{- 3 + 5}{2 \cdot 1}

数字相加/相减:

- 3 + 5 = 2

= \frac{2}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{2}{2}

使用法则

\frac{a}{a} = 1

= 1

v = \frac{- 3 - 5}{2 \cdot 1} : - 4

\frac{- 3 - 5}{2 \cdot 1}

数字相减:

- 3 - 5 = - 8

= \frac{- 8}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 8}{2}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{8}{2}

数字相除:

\frac{8}{2} = 4

= - 4

二次方程组的解是:

v = 1, v = - 4

v = 1, v = - 4

代回

v = u^{3}

，求解

u

解

u^{3} = 1 : u = 1, u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

u^{3} = 1

对于

x^{3} = f (a)

解为

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 1, u = \frac{- 1 + 3 i}{2}, u = \frac{- 1 - 3 i}{2}

化简

\frac{- 1 + 3 i}{2} : - \frac{1}{2} + i \frac{3}{2}

\frac{- 1 + 3 i}{2}

将

\frac{- 1 + 3 i}{2}

改写成标准复数形式:

- \frac{1}{2} + \frac{3}{2} i

\frac{- 1 + 3 i}{2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 + 3 i}{2} = - \frac{1}{2} + \frac{3 i}{2}

= - \frac{1}{2} + \frac{3 i}{2}

= - \frac{1}{2} + \frac{3}{2} i

化简

\frac{- 1 - 3 i}{2} : - \frac{1}{2} - i \frac{3}{2}

\frac{- 1 - 3 i}{2}

将

\frac{- 1 - 3 i}{2}

改写成标准复数形式:

- \frac{1}{2} - \frac{3}{2} i

\frac{- 1 - 3 i}{2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 - 3 i}{2} = - \frac{1}{2} - \frac{3 i}{2}

= - \frac{1}{2} - \frac{3 i}{2}

= - \frac{1}{2} - \frac{3}{2} i

u = 1, u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

解

u^{3} = - 4 : u = - 34, u = \frac{3 4}{2} - i \frac{3 4 3}{2}, u = \frac{3 4}{2} + i \frac{3 4 3}{2}

u^{3} = - 4

对于

x^{3} = f (a)

解为

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 3 - 4, u = 3 - 4 \frac{- 1 + 3 i}{2}, u = 3 - 4 \frac{- 1 - 3 i}{2}

3 - 4 = - 34

3 - 4

使用根式运算法则:

n - a = - n a,

若

n

是奇数

3 - 4 = - 34

= - 34

化简

3 - 4 \frac{- 1 + 3 i}{2} : \frac{3 4}{2} - i \frac{3 4 3}{2}

3 - 4 \frac{- 1 + 3 i}{2}

3 - 4 = - 34

3 - 4

使用根式运算法则:

n - a = - n a,

若

n

是奇数

3 - 4 = - 34

= - 34

= - 34 \frac{- 1 + 3 i}{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{( - 1 + 3 i ) 3 4}{2}

将

- \frac{( - 1 + 3 i ) 3 4}{2}

改写成标准复数形式:

\frac{3 4}{2} - \frac{3 4 3}{2} i

- \frac{( - 1 + 3 i ) 3 4}{2}

乘开

(- 1 + 3 i) 34 : - 34 + 343 i

(- 1 + 3 i) 34

= 34 (- 1 + 3 i)

使用分配律:

a (b + c) = ab + a c

a = 34, b = - 1, c = 3 i

= 34 (- 1) + 343 i

使用加减运算法则

+ (- a) = - a

= - 1 \cdot 34 + 343 i

乘以:

1 \cdot 34 = 34

= - 34 + 343 i

= - \frac{- 3 4 + 3 4 3 i}{2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 3 4 + 3 4 3 i}{2} = - (- \frac{3 4}{2}) - (\frac{3 4 3 i}{2})

= - (- \frac{3 4}{2}) - (\frac{3 4 3 i}{2})

去除括号:

(a) = a, - (- a) = a

= \frac{3 4}{2} - \frac{3 4 3 i}{2}

= \frac{3 4}{2} - \frac{3 4 3}{2} i

化简

3 - 4 \frac{- 1 - 3 i}{2} : \frac{3 4}{2} + i \frac{3 4 3}{2}

3 - 4 \frac{- 1 - 3 i}{2}

3 - 4 = - 34

3 - 4

使用根式运算法则:

n - a = - n a,

若

n

是奇数

3 - 4 = - 34

= - 34

= - 34 \frac{- 1 - 3 i}{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{( - 1 - 3 i ) 3 4}{2}

将

- \frac{( - 1 - 3 i ) 3 4}{2}

改写成标准复数形式:

\frac{3 4}{2} + \frac{3 4 3}{2} i

- \frac{( - 1 - 3 i ) 3 4}{2}

乘开

(- 1 - 3 i) 34 : - 34 - 343 i

(- 1 - 3 i) 34

= 34 (- 1 - 3 i)

使用分配律:

a (b - c) = ab - a c

a = 34, b = - 1, c = 3 i

= 34 (- 1) - 343 i

使用加减运算法则

+ (- a) = - a

= - 1 \cdot 34 - 343 i

乘以:

1 \cdot 34 = 34

= - 34 - 343 i

= - \frac{- 3 4 - 3 4 3 i}{2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 3 4 - 3 4 3 i}{2} = - (- \frac{3 4}{2}) - (- \frac{3 4 3 i}{2})

= - (- \frac{3 4}{2}) - (- \frac{3 4 3 i}{2})

使用法则

- (- a) = a

= \frac{3 4}{2} + \frac{3 4 3 i}{2}

= \frac{3 4}{2} + \frac{3 4 3}{2} i

u = - 34, u = \frac{3 4}{2} - i \frac{3 4 3}{2}, u = \frac{3 4}{2} + i \frac{3 4 3}{2}

解为

u = 1, u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}, u = - 34, u = \frac{3 4}{2} - i \frac{3 4 3}{2}, u = \frac{3 4}{2} + i \frac{3 4 3}{2}

u = cos (x)

代回

cos (x) = 1, cos (x) = - \frac{1}{2} + i \frac{3}{2}, cos (x) = - \frac{1}{2} - i \frac{3}{2}, cos (x) = - 34, cos (x) = \frac{3 4}{2} - i \frac{3 4 3}{2}, cos (x) = \frac{3 4}{2} + i \frac{3 4 3}{2}

cos (x) = 1, cos (x) = - \frac{1}{2} + i \frac{3}{2}, cos (x) = - \frac{1}{2} - i \frac{3}{2}, cos (x) = - 34, cos (x) = \frac{3 4}{2} - i \frac{3 4 3}{2}, cos (x) = \frac{3 4}{2} + i \frac{3 4 3}{2}

cos (x) = 1 : x = 2 πn

cos (x) = 1

cos (x) = 1

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - \frac{1}{2} + i \frac{3}{2} :

无解

cos (x) = - \frac{1}{2} + i \frac{3}{2}

无解

cos (x) = - \frac{1}{2} - i \frac{3}{2} :

无解

cos (x) = - \frac{1}{2} - i \frac{3}{2}

无解

cos (x) = - 34 :

无解

cos (x) = - 34

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{3 4}{2} - i \frac{3 4 3}{2} :

无解

cos (x) = \frac{3 4}{2} - i \frac{3 4 3}{2}

无解

cos (x) = \frac{3 4}{2} + i \frac{3 4 3}{2} :

无解

cos (x) = \frac{3 4}{2} + i \frac{3 4 3}{2}

无解

合并所有解

x = 2 πn

作图

流行的例子

- sin^{2} (x) = - 1

1+tan(x)=sec^2(x)

1 + tan (x) = sec^{2} (x)

4cosh(2x)=4+sinh(2x)

4 cosh (2 x) = 4 + sinh (2 x)

|sin(x)|=sin(x)+2

∣ sin (x) ∣ = sin (x) + 2

sin (a) = 0.2315